Giant component and vacant set for random walk on a discrete torus

Abstract

We consider random walk on a discrete torus $E$ of side-length $N$, in sufficiently high dimension $d$. We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time $u{N}^d$. We show that when $u$ is chosen small, as $N$ tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const $\log N$. Moreover, this connected component occupies a non-degenerate fraction of the total number of sites $N^d$ of $E$, and any point of $E$ lies within distance $N^{\beta }$ of this component, with $\beta$ an arbitrary positive number.